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Particle in 3D box

  1. Sep 9, 2006 #1
    This is the question:

    For a particle in a three-dimensional box of sides a, b, and c, where a does not equal b and b=c, make a table of n_x, n_y, and n_z, the energies, and the degeneracies of the levels in which the quantum numbers range from 0 to 4 (Take ((a^2)/(b^2)) = 2).

    Ok, I think I have an idea of what I'm supposed to do, but I'm a little confused on two parts of the question. When they ask for a table using quantum numbers from 0 to 4, would that mean I'd have to make a long list with n_x, n_y, and n_z like:
    0 0 0
    0 0 1
    0 0 2
    0 0 3
    0 0 4
    0 0 5

    and so on until I've listed all 125? Someone in my class said that there were 64 because of the zeroes, however I didn't really understand why we would be able to. I know that:
    [​IMG]
    So once I list all the states, I would have to substitute in the values for n_x, n_y, and n_z - but am I supposed to be able to get numerical values for the energies? I can get solve for a in terms of b and such, then plug in, but I still wouldn't get any numbers. Once I can figure out what I have to list, all I'd have to do to list the degeneracy is count the amount of states with different quantum numbers that have the same energy.
     
  2. jcsd
  3. Sep 10, 2006 #2
    I assume they mean [tex] n_x=1,2,3,4 n_y=1,2,3,4 n_z=1,2,3,4 [/tex]

    Since b=c you get 40 different cases, not 64, because for instance [tex] n_y=2,n_z=1 [/tex] and [tex] n_y=1,n_z=2 [/tex] are energy equivalent cases. You can combine [tex] n_y,n_z[/tex] in 10 different non degenerate ways(not 16) and then combine that with the 4 possible cases for [tex] n_x[/tex]

    n can not be zero thats probably what your friend mean. Consider a particle in a one dimensional box. If you put n=0 it basicly means there is no particle. the lowest state is when all three n values are =1.

    Numerical answeres doesnt seem to be possible but you can easily express them as multiples since [tex] \frac{a^2}{b^2}=2[/tex]

    Hope this helped a bit?
     
  4. Sep 10, 2006 #3
    I see what you mean now, although now I feel bad since I went ahead and listed 64 cases. At least now I can shorten my answer. Thanks a lot, this cleared up the bit about the zero.
     
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